Lie Bialgebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 21 March 2011

Lie bialgebras

1.1 A Lie bialgebra is a Lie algebra with bracket , and cobracket Ο†:π”€β†’π”€βŠ—π”€ such that
  1. π”€βˆ— with bracket Ο†βˆ—: π”€βˆ—βŠ—π”€βˆ—β†’ π”€βˆ— is a Lie algebra, and
  2. Ο† satisfies the 1-cocycle condition, Ο† x,y = xβŠ—1+1βŠ—x,Ο†y - yβŠ—1+1βŠ—y,Ο†x , for all x,yβˆˆπ”€.
In 1), Ο†βˆ—: π”€βŠ—π”€βˆ—β†’ π”€βˆ— is the induced mapping on the dual spaces. Given an element xβˆ—βˆˆπ”€βˆ— let us let ⟨xβˆ—,a⟩=xβˆ— a denote the evaluation of xβˆ— on the element aβˆˆπ”€. Then Ο†βˆ—: π”€βŠ—π”€βˆ—β†’ π”€βˆ— is given explicitly by βŸ¨Ο†βˆ—xβˆ—βŠ—yβˆ—,aβŠ—b⟩= ⟨xβˆ—,a⟩ ⟨yβˆ—,b⟩, for all xβˆ—,yβˆ—βˆˆπ”€βˆ— . Note that Ο†βˆ—:π”€βˆ—βŠ—π”€βˆ—β†’π”€βˆ— is a well defined map since π”€βˆ—βŠ—π”€βˆ—βŠ† π”€βŠ—π”€βˆ— (even if 𝔀 is infinite dimensional).

1.2 An element rβˆˆπ”€βŠ—π”€=V =C0𝔀,π”€βŠ—π”€ determines a 1-coboundary dr∈B1 𝔀,π”€βŠ—π”€ given by drx= 1βŠ—x+xβŠ—1,r , for all xβˆˆπ”€. Since d2=0 we know that every 1-coboundary is a 1-cocycle. Thus rβˆˆπ”€βŠ—π”€ determines a 1-cocycle and posibly a bialgebra structure on 𝔀. Thus it is a natural question:

  1. Given an element rβˆˆπ”€βŠ—π”€, what conditions should we place on r to guarantee that the map Ξ΄:π”€β†’π”€βŠ—π”€ determined by Ξ΄x= 1βŠ—x+xβŠ—1,r , determines a Lie bialgebra structure on 𝔀?
This question is answered by the following proposition:

Let 𝔀 be a Lie algebra, let rβˆˆπ”€βŠ—π”€ and define a map dr:π”€β†’π”€βŠ—π”€ by drx= 1βŠ—x+xβŠ—1,r . Let ρ= 12 r12-r21 and let P= 12 r12+r21 so that r=P+ρ.

  1. The map drβˆ—: π”€βˆ—βŠ—π”€βˆ—β†’ π”€βˆ— satisfies the skew-symmetric condition if and only if adβŠ—2 r12+r21 =0 for all xβˆˆπ”€.
  2. Assume that P is adβŠ—2 invariant. Then drβˆ— satisfies the Jacobi identity if and only if adβŠ—3 r12 ,r13+ r12r23+ r13r23 = 0 for all xβˆˆπ”€.

If r= βˆ‘ i aiβŠ—bi then the notation above is r12+r21= βˆ‘ i aiβŠ—bi+ biβŠ—ai , r12r13+ r12r23+ r13r23= βˆ‘ i,j aiajβŠ— biβŠ—bj+ aiβŠ—biaj βŠ—bj+ aiβŠ—ajβŠ— bibj . Condition a) states that r12+r21 is 𝔀-invariant and condition b) states that r12r13+ r12r23+ r13r23 is 𝔀-invariant.

1.4 A quasitraingular Lie bialgebra is a pair 𝔀,r such that 𝔀 is a Lie bialgebra, rβˆˆπ”€βŠ—π”€ , the cobraket in 𝔀 is equal to dr and r satisfies r12+r21=0, i.e, rβˆˆβ‹€2𝔀 .

A triangular Lie bialgebra is a pair 𝔀,r such that 𝔀 is a Lie bialgebra, rβˆˆπ”€βŠ—π”€, the cobracket in 𝔀 is equal to dr and r satisfies r12+r21=0,and r12r13+ r12r23+ r13r23=0. The equation r12r13+ r12r23+ r13r23 =0 is the classical Yang-Baxter equation (CYBE).

1.5 If 𝔀 is a semisimple Lie algebra over a field of characteristic 0, then it follows from Whitehead's lemma ([J] III Β§7 Lemma 3 p.77 and Thm. 13 p.95) that H1𝔀,r=0 for all finite dimensional 𝔀-modules V. Thus all 1-cocycles are coboundaries; in this case, if Ο†:π”€β†’π”€βŠ—π”€ is any linear map which satisfies the 1-cocycle condition then there is an element rβˆˆπ”€βŠ—π”€ such that Ο†x= 1βŠ—x+xβŠ—1,r for all xβˆˆπ”€.

Manin triples and the double

2.1 Let ⟨,⟩:π”­βŠ—π”­β†’π”­ be a symmetric bilinear form on a vector space 𝔭, i.e. ⟨x,y⟩=⟨y,x⟩ for all x,yβˆˆπ”­.

The form ⟨,⟩ is nondegenerate if the map given by βˆ—: 𝔭 β†’ π”­βˆ— x ↦ ⟨x,β‹…βŸ© is an isomorphism. Alternatively, the form ⟨,⟩ is invariant if for every xβˆˆπ”­, xβ‰ 0 there is a yβˆˆπ”­ such that ⟨x,yβŸ©β‰ 0. A third way to say it is that ⟨,⟩ is invariant if the null space of the form N= xβˆˆπ”­βˆ£βŸ¨x,y⟩=0  for all yβˆˆπ”­ is 0. A fourht way to say it is that the matrix of the form (with respect to any fixed basis of 𝔭) has nonzero determininant.

A subspace 𝔭′ of 𝔭 is isotropic if ⟨x,x⟩=0 for all xβˆˆπ”­β€².

Given vector spaces 𝔭1 and 𝔭2 with symmetric bilinear forms ⟨,⟩1 and ⟨,⟩2 respectively then the vector space 𝔭1βŠ—π”­2 has a bilinear form ⟨,⟩ given by ⟨xβŠ—y,zβŠ—w⟩= ⟨x,z⟩1 ⟨y,w⟩2 for all x,zβˆˆπ”­1 and y,wβˆˆπ”­2. In particular, if 𝔭 is a vector space with a bilinear form ⟨,⟩ then π”­βŠ—π”­ has a bilinear form given by ⟨xβŠ—y,zβŠ—w⟩= ⟨x,z⟩⟨y,w⟩ for all x,y,z,wβˆˆπ”­.

Suppose that the vector space 𝔭 is a Lie algebra. The form ⟨,⟩ is invariant if ⟨adzx,y⟩= -⟨x,adzy⟩, for all x,y,zβˆˆπ”­.

2.2 A Manin triple is a triple 𝔭,𝔭1,𝔭2 such that

  1. 𝔭 is a Lie algebra with a nondegenerat invariant symmetric bilinear form ⟨,⟩ and
  2. 𝔭1 and 𝔭2 are isotropic Lie subalgebra of 𝔭.
  3. 𝔭=𝔭1βŠ•π”­2 as vector spaces.

Let 𝔭,𝔭1,𝔭2 be a Manin triple. Then 𝔭1 is a Lie algebra with cobracket Ξ΄:𝔭1→𝔭1βŠ•π”­2 determined by the equation ⟨δx ,y1βŠ—y2⟩= ⟨x,y1,y2⟩, for all xβˆˆπ”­1 and all y1,y2βˆˆπ”­2 . In other words, Ξ΄ is the adjoint of the bracket on 𝔭2.

Let 𝔀δ be a Lie bialgebra. Then the triple π”€βŠ•π”€βˆ—,𝔀,π”€βˆ— is a Manin triple where

  1. The bilinear form ⟨,⟩ on π”€βŠ•π”€βˆ— is given by ⟨x1+ y1βˆ— ,x2+ y2βˆ— ⟩ = ⟨x1, y2βˆ— ⟩+ ⟨x2, y1βˆ— ⟩ = y2βˆ— x1+ y1βˆ— x2, for all x1,x2βˆˆπ”€ and all y1βˆ— , y2βˆ— βˆˆπ”€βˆ— , and
  2. The bracket on π”€βŠ•π”€βˆ— is determined by the formulas ⟨ y1βˆ— y2βˆ— ,p⟩ = ⟨ y1βˆ— βŠ— y2βˆ— , Ξ΄p ⟩, if pβˆˆπ”€, 0, if pβˆˆπ”€βˆ—, ⟨xyβˆ—,p⟩ = ⟨yβˆ—,px⟩, if pβˆˆπ”€, ⟨x,yβˆ—p⟩, if pβˆˆπ”€βˆ—, where xβˆˆπ”€ and y,y1, y2βˆ— βˆˆπ”€βˆ— .

This is essentially the only way to define things so that the bracket on π”€βˆ— is the dual of the cobracket on 𝔀 and so that the bilinear form is invariant. The Lie algebra π”€βŠ•π”€=Dg constructed from the the Lie bialgebra 𝔀 is called the double corresponding to the Lie algebra 𝔀.

The previous two propositions show that there is a one-to-one correspondence between Lie bialgebras and Manin triples.

Let 𝔀 be a Lie bialgebra and D𝔀=π”€βŠ•π”€βˆ— be the double of 𝔀. Let ai be a basis of 𝔀 and let ai be the dual basis in π”€βˆ—. Define r= βˆ‘ i aiβŠ—ai∈ π”€βŠ—π”€βˆ— βŠ† Dπ”€βŠ—D𝔀. Then

  1. The element r does not depend on the choice of basis ai of 𝔀.
  2. r satisfies the CYBE.
  3. D𝔀,r is a quasitriangular Lie bialgebra.

Proofs

Let 𝔀 be a Lie algebra, let rβˆˆπ”€βŠ—π”€ and define a map dr:π”€β†’π”€βŠ—π”€ by drx= 1βŠ—x+xβŠ—1,r . Let ρ= 12 r12-r21 and let P= 12 r12-r21 so that r=P+ρ.

  1. The map drβˆ—: π”€βˆ—βŠ—π”€βˆ—β†’ π”€βˆ— satisfies the skew-symmetric condition if and only if adβŠ—2 r12+r21 =0 for all xβˆˆπ”€.
  2. Assume that P is adβŠ—2 invariant. Then drβˆ— satisfies the Jacobi identity if and only if adβŠ—3 r12 ,r13+ r12r23+ r13r23 = 0 for all xβˆˆπ”€.

Proof:

Let 𝔭,𝔭1,𝔭2 be a Manin triple. Then 𝔭1 is a Lie bialgebra with cobracket Ξ΄:𝔭1β†’ 𝔭1βŠ—π”­1 determined by the equation ⟨δx, y1βŠ—y2 ⟩= ⟨x, y1,y2⟩ for all xβˆˆπ”­1 and all y1,y2βˆˆπ”­2. In other words, Ξ΄ is the adjoint of the bracket on 𝔭2.

Proof:

Let 𝔀,Ξ΄ be a Lie bialgebra. Then the triple π”€βŠ•π”€βˆ—,𝔀,π”€βˆ— is a Manin triple where

  1. The bilinear form ⟨,⟩ on π”€βŠ•π”€βˆ— is given by ⟨ x1+ y1βˆ— , x2+ y2βˆ— ⟩ = ⟨ x1 , y2βˆ— ⟩ + ⟨ x2 , y1βˆ— ⟩ = y2βˆ— x1+ y1βˆ— x2, for all x1,x2βˆˆπ”€ and all y1βˆ— , y2βˆ— βˆˆπ”€βˆ—
  2. The bracket on π”€βŠ•π”€βˆ— is determined by the formulas ⟨ y1βˆ— y2βˆ— ,p⟩ = ⟨ y1βˆ— βŠ— y2βˆ— , Ξ΄p ⟩, if pβˆˆπ”€, 0, if pβˆˆπ”€βˆ—, ⟨xyβˆ—,p⟩ = ⟨yβˆ—,px⟩, if pβˆˆπ”€, ⟨x,yβˆ—p⟩, if pβˆˆπ”€βˆ—, where xβˆˆπ”€ and y,y1, y2βˆ— βˆˆπ”€βˆ— .

Proof:

Let 𝔀 be a Lie bialgebra and let D𝔀=π”€βŠ•π”€βˆ— be the double of 𝔀. Let ai be a basis of 𝔀 and ai be the dual basis in π”€βˆ—. Define r= βˆ‘ i aiβŠ—ai∈ π”€βŠ—π”€βˆ—βŠ† Dπ”€βŠ—D𝔀. Then

  1. The element r does not depend on the choice of basis ai of 𝔀.
  2. r satisfies the CYBE.
  3. D𝔀,r is a quasitriangular Lie bialgebra.

Proof:

References

The motivating reference is

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

There is a detailed exposition of Lie bialgebras in the following article

[DHL] H.-D. Doebner, Hennig, J. D. and W. LΓΌcke, Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989), Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63. MR1201823

[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.

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